Loop Induced Single Top Partner Production and Decay at the LHC

Most searches for top partners, $T$, are concerned with top partner pair production. However, as these bounds become increasingly stringent, the LHC energy will saturate and single top partner production will become more important. In this paper we study the novel signature of the top partner produced in association with the SM top, $pp\rightarrow T\overline{t}+t\overline{T}$, in a model where the Standard Model (SM) is extended by a vector-like $SU(2)_L$ singlet fermion top partner and a real, SM gauge singlet scalar, $S$. In this model, $pp\rightarrow T\overline{t}+t\overline{T}$ production is possible through loops mediated by the scalar singlet. We find that, with reasonable coupling strengths, the production rate of this channel can dominate top partner pair production at top partner masses of $m_T\gtrsim 1.5$ TeV. In addition, this model allows for the exotic decay modes $T\rightarrow tg$, $T\rightarrow t \gamma$, and $T\rightarrow t S$. In much of the parameter space the loop induced decay $T\rightarrow tg$ dominates and the top partner is quite long lived. New search strategies are necessary to cover these decay modes. We project the the sensitivity of the high luminosity LHC to $pp\rightarrow T\overline{t}+t\overline{T}$ via a realistic collider study. We find with 3 ab$^{-1}$, the LHC is sensitive to this process for masses $m_T\lesssim2$ TeV. In addition, we provide appendices detailing the renormalization of this model.


Introduction
The Large Hadron Collider (LHC) is quickly accumulating data at the energy frontier of particle physics. While the the LHC is searching for many types of beyond-the-Standard Model (BSM) physics, of particular interest are searches for partners of the SM top quark. In many models that solve the naturalness problem, top quark partners are postulated to exist and cancel the quadratic corrections to the Higgs mass, stabilizing the Higgs at the partner becomes quite long lived, necessitating an update of search strategies.
In this paper we study a simplified model containing a top partner and a real, SM gauge singlet scalar. We will show that this model has interesting signatures and that LHC is sensitive to new regions of parameter space via pp → T t + tT production. In Section 2 we introduce the model and couplings of the new particles. The production and decay rates of the top partner are studied in Section 3, and the production and decay rates and scalar are studied in Section 4. Current experimental constraints on top partners and scalar singlets are presented in Section 5. In Section 6, we perform a realistic collider study for the process pp → T t + tT → ttS → ttgg. We conclude in Section 7. In addition, we attach three appendices with necessary calculation details. In Appendix A we present the details of the wave-function and mass renormalization of the top sector. Vertex counterterms for T − t − g, T − t − γ, and T − t − Z are presented in Appendix B. In Appendix C we give the parameterization of energy smearing for the collider study.

The Model
We consider a model consisting of a vector-like SU (2) L singlet top partner, T 2 , and a real SM gauge singlet scalar S. A similar model has been consider in Ref. [58]. For simplicity and to avoid flavor constraints, the top partner is only allowed to couple to the third generation SM quarks: The allowed Yukawa interactions and mass terms are + λ 1 ST 2L T 1R + λ 2 ST 2L T 2R + h.c., (2.2) where Φ is the SM Higgs doublet, Φ = iσ 2 Φ * , and σ 2 is a Pauli matrix. The most general renormalizable scalar potential has the form [38] V (Φ, After EW symmetry breaking (EWSB), in general both the scalar S and Higgs doublet Φ can develop vacuum expectation values (vevs): Φ T = (0, v/ √ 2) and S = x where v = 246 GeV is the SM Higgs doublet vev. Since S is a gauge singlet and there are no discrete symmetries imposed, shifting to the vacuum S = x + s is a field redefinition that leaves all the symmetries intact. Hence, it is unphysical and we are free to choose x = 0 [38]. Two possible ways to understand this are: (1) All possible interaction terms of S are already contained in the scalar potential and Yukawa interactions, Eqs. (2.2) and (2.3). Hence, shifting to the vacuum S = x + s does not introduce any new interactions and is unphysical. (2) After S obtains a vev, any discrete symmetry that S has is broken and all interactions in Eqs. (2.2) and (2.3) are possible. Hence, the scalar S can be interpreted as the field after already shifting to the vacuum with x = 0.
Also after EWSB, it is possible for the scalar S and Higgs boson h to mix. However, since the focus of this paper is the production and decay of the top partner, for simplicity we set the scalar mixing angle to zero. This is equivalent to setting a 1 = 0 in Eq. (2.3). Hence, h and S are mass eigenstates with masses m h = 125 GeV [79][80][81] and m S , respectively; such that h is the observed Higgs boson [82,83].
There is another possible simplification of the Lagrangian. Since T 2R and T 1R have the same quantum numbers and T 2L and T 2R are two different Weyl-spinors, the off-diagonal vector-like mass-term, M 12 , can be removed via the field redefinitions [84] The Yukawa interactions and mass terms are then For simplicity, we assume all couplings are real. The relevant kinetic terms are then where the covariant derivatives are where σ a are Pauli matrices and T A are the fundamental SU (3) representation matrices.

Scalar Couplings to Top Partners
After EWSB, in the unitary gauge Φ = (0, (h + v)/ √ 2) T the quark masses and Yukawa interactions are (2.9) where the top quark and partner are χ τ = t 1τ t 2τ (2.10) with τ = L, R, and the mass and Yukawa matrices are y t λ t 0 0 , and Y S = 0 0 λ 1 λ 2 . (2.11) The top-quark mass matrix can be diagonalized via the bi-unitary transformation t 1τ t 2τ = cos θ τ sin θ τ − sin θ τ cos θ τ t τ T τ . (2.12) The mass eigenstates are t and T with masses m t = 173 GeV [85] and m T , respectively, such that t is the observed SM-like top quark. Upon diagonalization, the Higgs Yukawa coupling, y t , λ t , and the vector like mass M 2 can be expressed in terms of the mixing angle θ L and masses m t , m T : Additionally, only one of the mixing angles θ L and θ R is free: (2.14) The independent parameters of this theory are then θ L , m T , m S , λ 1 , and λ 2 .
(2. 15) After rotating to the mass eigenbasis, the quark masses and scalar couplings are −L Y uk = h λ h tt tt + λ h T T T T + t λ h tT P R + λ h T t P L T + T λ h T t P R + λ h tT P L t + S λ S tt tt + λ S T T T T + t λ S tT P R + λ S T t P L T + T λ S T t P R + λ S tT P L t + m t tt + m T T T + m b 1 + h v bb, (2.16) where m b = y b v/ √ 2 is the bottom quark mass, the Higgs boson couplings are

Z and W ± Couplings to Top Partners
After diagonalizing the top quark mass matrix, the Z and W couplings to the third generation and top partner are altered as well as introducing the flavor off diagonal coupling t − T − Z. The interactions relevant for our analysis are where c W = cos θ W , s W = sin θ W , θ W is the weak mixing angle, g is the weak coupling constant, g Z L = 1 2 − 2 3 s 2 W , and g Z R = − 2 3 s 2 W . Since electromagnetism and SU (3) are unbroken, the top quark and partner just couple to photons and gluons according to their electric and color charges. We use the Z-mass, the Fermi decay constant, and the electric coupling at the Z-pole as input parameters [85]: m Z = 91.1876 GeV, G F = 1.16637 × 10 −5 GeV −2 , α(m Z ) −1 = 127.9. (2.20) The other EW parameters (g, θ W , v, m W ) are calculated using the tree level relations where m W is the W -mass.

Effective Field Theory
In the limit that m S m T , v, the scalar S can be integrated out. The lowest dimension operators that contribute to top partner production and decay are the dipole operators: (2.22) where the hypercharge and gluon field strength tensors are and f ABC is the SU (3) structure constant. These interactions arise from the processes shown in Fig. 1. Taking the limit that m S m T and that EW symmetry is restored (v → 0, sin θ L → 0), we calculate T → tB and T → tg. The details of the necessary renormalization counterterms can be found in Appendices A and B. Matching onto the EFT, we find the Wilson coefficients: Note that the ratio of the Wilson coefficients c G /c B = 3 g s c W /(2 e) is completely determined by the the ratio of the strong and Hypercharge coupling constants. This is because the structure of the loop diagrams in Fig. 1 are essentially the same with the only difference being the external gauge boson and their couplings to the top partner. Also, although the operators in Eq. (2.22) are dimension five, the Wilson coefficients are suppressed by two powers of m S (m T /m 2 S ) and not one power (1/m S ). The dipole operators couple left-and right-chiral fields. Hence, the loop diagram needs an odd number of changes in chirality. From just the couplings, the diagrams in Fig. 1 have an even number of chiral flips. An additional mass insertion is needed and one power of m T in the numerator is necessary. The operators are then suppressed by m T /m 2 S and not 1/m S .

Production and Decay of Top Partner
We now discuss the production and decay of the top partner, T , in the model presented in Sec. 2. To produce the the numerical results we implement the model in FeynArts [86] via FeynRules [87,88]. Matrix element squareds are then generated with FormCalc [89]. We use the NNPDF2.3QED [90] parton distribution functions (pdfs) as implemented in LHAPDF6 [91]. We also use the strong coupling constant as implemented in LHAPDF6. Details on the wave-function renormalization and vertex counterterms needed for the calculations in this section can be found in the Appendices A and B.

Top Partner Production Channels
There are many possible production channels for top partners. Figure 2 shows the classic tree level mechanisms: (a-c) top partner pair production (T T ), (d,e) top partner plus jet production (T +jet), and (f,g) top partner plus W ± production (T W ) 1 . We collectively refer to final states with a single T produced in association with a SM particle as single top partner production. Although top partner pair production is dominant for much of the parameter region, single top partner plus jet production can become important for very massive T despite the b-quark pdf suppression [16][17][18][19][20][21][22]58]. This is mainly due to two Figure 2. Standard production modes of top partners at the LHC for (a-c) pair production, (d,e) top partner plus jet production, and (f,g) top partner plus W − production. There are conjugate processes for (d-g) that are not shown here.  Figure 3. Production cross sections at the √ S = 14 TeV LHC for (green dashed) top partner pair production, (black solid) top partner production in association with a top quark, (red dashdash-dot) top partner plus jet production, and (blue dotted) top partner plus W ± production. The parameters are set at a scalar mass m S = 200 GeV, couplings λ 1 = λ 2 = 3, and mixing angles (a) sin θ L = 0.15 and (b) sin θ L = 0.01. Factorization, µ f , and renormalization, µ r , scales are set to the sum of the final state particle masses. effects: the gluon pdf drops precipitously at high mass suppressing the T T rate and top partner pair production starts saturating the available LHC phase space at high energies. This can be clearly seen in Fig. 3, which compares the cross sections of various top partner production modes as a function of the top partner mass m T . At the √ S = 14 TeV LHC and for a mixing angle of sin θ L = 0.15, Fig. 3(a), the T +jet production becomes larger than that of top partner pair production at a mass around m T ∼ 700 GeV and T W production Figure 4. Representative Feynman diagrams for single production of T in association with a top quark for (a) tree level through an s-channel Z, (b-c) quark-antiquark initial state, and (d-i) gluon fusion. We have not shown the conjugate process, counterterms, off-diagonal self energies of the external top and top partner, or any loops with internal Goldstone bosons, Z, or W ± . is comparable to T T production for m T ∼ 2.5 TeV.
However, for the simplest model where the SM is augmented by a single SU (2) L singlet top partner, single top partner production relies on the b − W − T coupling. This coupling is proportional to the to the T − t mixing angle sin θ L , as can be seen in Eq. (2.19). Hence, the production cross section is proportional to sin 2 θ L and vanishes as the mixing angle goes to zero. In fact, as shown in Fig. 3(b), T T always dominates T +jet and T W for sin θ L = 0.01 at the √ S = 14 TeV LHC for all masses shown. In the model presented in Sec. 2, in addition to the production modes in Fig. 2, the flavor-off diagonal couplings between the new scalar, top partner, and top quark introduces new loop level production mode: top partner production in association with a top quark (T t). Representative Feynman diagrams with flavor off-diagonal scalar couplings for this process are show in Fig. 4. We do not show the conjugate process; counterterm diagrams; diagrams with Goldstone bosons, Zs, or W ± s internal to the loop; or external off-diagonal self-energy diagrams between the top quark and top partner. However, these are included in the calculation. Although T t production is allowed at tree level for non-zero sin θ L , as with T +jet and T W production, the tree level T t cross section is proportional to sin 2 θ L . Hence, it vanishes as sin θ L vanishes. However, the S − t − T and S − T − T couplings do  For m S > m T +m t , it is possible for the scalar to resonantly decay into the top partner and top through the diagram in Fig. 4(d). If the scalar is not too heavy, it will be possible to produce it and look for this decay channel at the LHC. This type of signal has been much studied and searched for [68,[92][93][94][95][96]. However, if the scalar is too heavy it will not be possible to produce it at the LHC. In this case, the EFT presented in Sec. 2.3 is relevant. As can be clearly seen, the production cross section is then suppressed by 1/m 4 S . For large scalar masses it is always negligible compared to pair production. Hence, for our discussion of T production we focus on the scenario where m S < m T + m t . However, as we will see, for m S m T the decay channels of the top partner are interesting and present a new phenomenology.
The importance of T t production can be seen in Fig. 3. For m S = 200 GeV and both sin θ L = 0.15 and sin θ L = 0.01, at the √ S = 14 TeV LHC the top partner plus top production rate is greater than that of top partner pair production for m T 1.5 TeV. While for sin θ L = 0.15, T +jet production is consistently larger than T t production, the situation changes drastically for smaller mixing angles. As can be seen by comparing Figs. 3(a) and 3(b), the T t rate does not greatly decrease as sin θ L becomes small. Figure 3(b) shows that T t is the dominant single top partner production mechanism for small mixing angles.
In Fig. 5 we show contours of LHC cross sections for top partner plus top production in λ 1 − λ 2 plane in the zero mixing sin θ L = 0 limit. This is presented for both m T = 1.5 TeV, noting that in the zero mixing limit the T t production rate is proportional to the coupling constants squared: Hence, contours of constant cross section correspond to |λ 1 | ∝ |λ 2 | −1 . For comparison, we also show the top production pair production rate (red dashed lines). As can be seen, there is a significant amount of parameter space for which the T t rate dominates T T . Using the simple relation in Eq. (3.1), for sin θ L = 0 and m S = 200 GeV, we find that at the √ S = 14 TeV LHC the T t cross section is larger than the T T cross section for |λ 1 λ 2 | 2.9 for m T = 1.5 GeV and In Fig. 6(a) we show various single top partner production rates as a function of sin θ L for m T = 1.5 TeV and m S = 200 GeV. At small mixing angles all the single top partner rates vanish except T t. It is expected that searches for T +jet production will limit sin θ L 0.02 − 0.06 [20]. Hence, this is the parameter region where top partner plus top production is most important. Also, for larger coupling constants λ 1,2 , the T t rate has little dependence on sin θ L , while for smaller λ 1,2 the dependence is stronger. This can be understood by
noting that for non-zero mixing angles, loop diagrams involving the Higgs, Z boson, W boson, and Goldstone bosons contribute to T t. For smaller λ 1,2 these contributions can compete with the scalar S contributions, introducing more sin θ L dependence. For larger λ 1,2 , the scalar S loops always dominate and mixing angle dependence is milder.
The dependence of the T t production rate on the scalar mass is shown in Fig. 6(b) for λ 1,2 = 3 and m T = 1.5 TeV. For all mixing angles, the cross section is larger for smaller scalar mass. The dependence of the cross section on m S does not change greatly for different sin θ L . Table 1 summarizes the results of top partner production with m S = 200 GeV. The left column gives parameter regions for which T t production is the dominant single top partner production mode. The right column gives parameter regions for which T t production dominates T T double production. For small mixing angles, T t is the dominant single top production mode, while T t production dominates T T production at large m T . Also, T t production is maximized for smaller scalar masses. Figure 7 shows representative Feynman diagrams for top partner decays. Searches for top partners typically rely on the T → th, T → tZ, and T → bW decays [69][70][71][72] as shown in Fig. 7(a)-7(c). However, in the model presented in Sec. 2, new top partner decay modes are available. For small enough scalar masses, m S + m t < m T , there is a new tree level decay T → tS, as shown in Fig. 7(d). Additionally, there are possible loop level decays, shown in Figs. 7(e)-7(g), that are important when the T → tS channel is kinematically forbidden and, as we will see, for sufficiently small mixing angle sin θ L . These new decay channels can change search strategies for fermionic top partners.

Top Partner Decay Channels
Again, in the loop diagrams in Fig. 7, we do not show external self-energies, external vacuum polarizations, loops with Z bosons, loops with W ± bosons, or loops with Goldstone bosons, although they are included in the calculations. Additionally, we only consider the leading contributions to each decay channel. Hence, T → tγ and T → tg are calculated at one loop. For T → th and T → bW , we only consider tree level decays. While there are loop corrections, for T → th they will be dependent on λ S tt , λ h T T , λ h tT , λ h T t , W − b − T , or the Z − t − T couplings in Eqs. (2.17-2.19), all of which are proportional to sin θ L . There is also  (2.17-2.19), which are also proportional to sin θ L . Since both the tree and one loop level contributions to T → th and T → W b are always proportional to sin θ L , we expect the tree level diagrams to dominate throughout parameter space and we do not calculate the loop contributions to these decays. The decay T → tZ is more complicated. The tree level component vanishes as sin θ L → 0. However, the loop contribution in Fig. 7(f) does not vanish as sin θ L → 0 since it depends on the λ S T t , λ S T T , and the Z − T − T couplings in Eqs. (2.18,2.19) which are non-zero for sin θ L = 0. Hence, we calculate tree and loop level diagrams to T → tZ so that the dominant contributions are included for all sin θ L .
We first consider m S + m t < m T , where T → tS is available. Figure 8 illustrates how the branching ratios of T depend on the top partner mass and mixing angle. Although not shown, we also calculated the branching ratios of T → tγ and T → tg, but they are negligible in this regime. As can be seen in Fig. 8(a), for smaller top partner masses the T → tS decay dominates while for larger masses the standard decays T → bW , T → tZ, and T → th dominate. This can be understood by considering the partial widths in the m T v, m S limit and counting sin The decays into SM final states dominate since the partial widths scale as m 3 T and the partial width Γ(T → tS) grows as m T . This can be understood via the Goldstone Equivalence Theorem and that the W, Z, h couplings are proportional to mass for very heavy m T . In fact, the SM decays obey the g/ / /Z g/ /Z/Z T T T S Figure 9. The scalar S decays into gg, γγ, γZ and ZZ final states through T loop in the zeromixing limit (sin θ L → 0).
Of course, allowing mixing between the scalar and Higgs boson will slightly complicate this scenario, since the two mass eigenstate scalars will be superpositions of the gauge singlet scalar and Higgs boson. Since the scalar would then have a component of the Higgs doublet, the parametric dependence of the widths is where θ is the scalar mixing angle and m T v, m S . Then Γ(T → tS) has a component that grows as m 3 T , but is suppressed by the scalar mixing angle. For simplicity, we are focusing on the scenario where the scalar mixing angle is zero, although, as is clear from Eq. (3.7), the precise phenomenology will change for non-zero scalar mixing [58]. However, while the branching ratios of the top partner can change, there are no new decay channels for the top partner in the non-zero scalar mixing scenario. Hence, we still capture the major phenomenological aspects of this model.
Precisely when the SM final states dominate will also depend on the coupling constants λ 1,2 and mixing angle sin θ L . In Fig. 8(b) we show the dependence of the top partner branching ratios on sin θ L for m S = 200 GeV and m T = 1.5 TeV. For larger mixing angles | sin θ L | 0.1−0.12, the decay into bottom quark and W dominates. However, as expected for sin θ L ∼ 0 the branching ratio of T → tS is very nearly 100% since the other tree level decay modes vanish.
When T → tS dominates, search strategies will strongly depend on the decay of the scalar. If S is allowed to have non-negligible mixing with the Higgs boson, the scalar will decay like a heavy Higgs boson. That is, we would expect S → W W , S → ZZ, S → tt, and S → hh to be tree level and dominate when they are allowed [58]. If λ 2 = 0, as-well-as , it is possible to apply a Z 2 symmetry on the top partner and scalar, T → −T and S → −S, while the SM fields are even SM → SM . The only available decay mode is then T → tS and the scalar S is a possible dark matter candidate.  Top partners are then pair produced and the signal is TT → tt + / E T [100]. The scenario we consider has no Z 2 symmetry and sets the scalar-Higgs mixing to zero. Then the only decay channels available to the scalar S are through loops of top quarks and top partners.
Depending on the precise mass of the scalar, the decays S → W W , S → ZZ, S → γγ, S → Zγ, S → hh, and S → gg will be possible. The S → hh and S → W W decay rates are mixing angle suppressed since all contributing diagrams are dependent on Hence, in the sin θ L = 0 limit, the scalar S decays to neutral gauge bosons, as shown in Fig. 9, and the branching ratios are determined by the gauge couplings. Then S → gg and T → tS → tgg are by far the dominate decay modes.

m S > m T − m t and Long Lived Top Partners
In Fig. 10 we show the (a,b) total widths and (c,d) branching ratios as a function of mixing angle sin θ L for scalar masses larger than top partner mass. The top partner mass is m T = 1.5 TeV, the scalar masses and couplings are (a,c) m S = 5 TeV, λ 1,2 = 1 and (b,d) m S = 10 TeV, λ 1,2 = 3. For mixing angle sin θ L 10 −4 − 10 −3 , the tree level decays dominate and the partial widths are independent of the scalar mass and couplings. For sin θ L 10 −3 − 10 −4 the loop level decay T → tg is the main mode. To determine the relative importance of the loop contributions it is useful to look at the T → tZ decay channel, since it is the only one for which we include both loop and tree level contributions. At sin θ L ∼ 10 −5 − 10 −4 there is a clear transition between a Γ(T → tZ) ∼ sin 2 θ L dependence expected at tree level and a width Γ(T → tZ) that is relatively independent of sin θ L . This is the passage between tree level and loop level dominance in T → tZ.
The dependence of Γ(T → tg), Γ(T → tγ), and Γ(T → tZ) on the model parameters at small angles can be understood by noting that for sin θ L ≈ 0, m T v, and m S m T , the EFT is Eq. (2.22) is valid. In this EFT, the partial widths are (3.10) Hence, the partial widths are independent of sin θ L and all have the same parametric dependence on the top partner mass, scalar mass, and couplings λ 1,2 . The branching ratios of the top partner in the m S > m T − m t regime are shown in Figs. 10(c) and 10(d). Although the values of the partial widths depend on the precise model parameters, the branching ratios are largely independent of model parameters for larger sin θ L 10 −3 or small sin θ L 10 −5 . The behavior of the branching ratios for sin θ L ∼ 10 −5 − 10 −3 depends on the relative dominance of the tree level and loop level contribution, and hence the model parameters, as discussed above. For mixing angles sin θ L 10 −3 , the tree level decays in to SM EW bosons T → bW , T → tZ and T → th dominate and they obey the expected relation BR(T → bW ) ≈ 2 BR(T → tZ) ≈ 2 BR(T → th) ≈ 50%. This can be understood by noting that in the heavy top partner regime, these partial widths only depend on sin θ L and m T and this dependence cancels in the ratios of the widths in Eqs. (3.4-3.6).
For sin θ L 10 −4 the decay T → tg dominates, while for sin θ L 10 −5 all loop level decays dominate and the branching ratios are approximately independent of the model parameters. For the EFT, since the partial widths in Eqs. (3.8-3.10) have the same parametric dependence, the branching ratios are independent of couplings λ 1,2 and masses m T , m S . Hence, the branching ratios are largely determined by the gauge coupling constants and weak mixing angle. Then the decay T → tg is by far the dominate mode due to the strong coupling constant. There are additional corrections from the Higgs vev to Eq.   from neglected dimension-6 operators of the form This can explain the O(10%) differences between the branching ratios at m S = 5 TeV and 10 TeV, as observed in Figs. 10(c) and 10(d).
For heavy scalars m S > m T −m t and zero mixing angle sin θ L = 0, Fig. 11 shows (a) the total width and (b) the decay length of the top partner for various parameter points. If the decay width of a colored particle is less than Λ QCD ∼ 100 MeV [85], we expect the particle to hadronize and bind with light quarks before it decays. As can be clearly seen, when the loop level decays of the top partner are dominant, we have the total width Γ T < Λ QCD for the vast majority of parameter space. Hence, the top partner almost always hadronizes before it decays. See for example Ref. [101] for a discussion of the phenomenology of top partner hadrons.
At threshold it may be possible for pair produced top partners to bind and form exotic heavy quarkonia, η T = T T . This will be possible if the decay widths of T and η T are less than the binding energy, E b , of η T . If this condition is not satisfied, the lifetime of η T will be less than the characteristic orbital time of the constituents and η T will not be a resonance. Assuming that the binding force is essentially Coulombic, this condition is [102,103]: where Γ η T is the η T decay width. The precise decay pattern of the exotic quarkonia depend on the model parameters. In addition to top partner decays, η T has decays into other SM final states. The dominant mode is η T → gg [101,103] with partial width Γ(η T → gg) ∼ 1 − 10 MeV [103,104]. Hence, if Γ T Γ(η T → gg) the condition to form quarkonia in Eq. (3.12) is always satisfied. Additionally, we have BR(η T → gg) ≈ 1 and can employ a typical search for exotic quarkonia [101,[103][104][105]. However, if |E b | Γ T Γ(η T → gg), the top partner decays are expected to dominate the η T decays. The top partners will decay according to the branching ratios in Fig. 10. For the parameter ranges in Fig. 11(a), the condition in Eq. (3.12) is always satisfied.
As can be seen in Fig. 11(b), for not too small couplings, the decay lengths of the top partners can be significant on the scales of collider experiments. This leads to many exotic phenomena such as displaced vertices [106][107][108], stopped particles [109][110][111], and long lived particles [112]. The different decay lengths can be categorized as it can be reconstructed as a displaced vertex offset from the primary vertex of the proton-proton interaction [108,[113][114][115][116][117][118]. The top partner has these decay lengths for the following parameter regions: • "Stable" particles: It is possible for charged and colored particles to be stable on collider scales [112]. Searches typically look for either high energy deposits in the trackers, measure time of flight with the muon systems, or search for decays in the hadronic calorimeter [119][120][121][122][123][124][125]. These searches are sensitive to decay lengths of O(1 m) − O(10 m) or longer. For both m S = 2.5 TeV and 10 TeV, top partners have these decay lengths for λ 1,2 10 −3 and m T 2 TeV. For m S = 10 TeV, top partners also have these decay lengths for λ 1,2 ∼ 10 −2 and m T 400 GeV.
• Stopped particles: Long lived colored particles hadronize and interact with the detectors, losing energy through ionization [109,110]. It is possible for all the energy to be lost and the particles to stop inside the hadronic calorimeter [110,111]. For O(1 TeV) colored particles, nearly 100% with speeds below β ∼ 0.25 − 0.3 will stop [110]. If the particle's lifetime is O(100 ns), they can be searched for as decays inside the hadronic calorimeter that are out of time with the bunch crossing [111,[126][127][128]. For particles to stop in the calorimeter, they must be long lived on collider time scales. Hence, much the same parameter space that gives "stable" particles gives stopped particles.

Summary
In Fig. 12 we show the (a) total width and (b) width to mass ratio for (black solid) m S = 200 GeV and (red dashed) m S = 10 TeV with sin θ L = 0.15. For m S > m T − m t , the T → St decay mode is no longer allowed. However, for non-negligible mixing angle, the tree level T → W b, T → tZ, and T → th are still available and growing as m 3 T . The result of decoupling S is to suppress the total width by ∼ 50% for m T ∼ 500 GeV and ∼ 10% for m T ∼ 2.5 TeV. For both cases, although the width to Higgs and gauge bosons increases with m 3 T , the width to mass ratio never exceeds 10% and can always be safely regarded as narrow.
We summarize our results for top partner decays in Tables 2 and 3. The possible ranges of the dominant top partner decay modes for different mixing angle and scalar mass ranges are shown in Table 2. In Table 3, we give representative parameter regions that give various collider signatures of long lived top partners.

Production and Decay of the Scalar
We now discuss the production and decay of the scalar, S, in the model presented in Sec. 2. We focus on the region of parameter space for which the scalar can be produced at the LHC  with reasonable rates, i.e. m S ∼ 100s GeV and m T > m S . As mentioned in the previous section, the scalar can be produced via decays of the top partner. The scalar can also be directly produced through gluon fusion mediated by top quark and top partner loops, similar to the loops in Fig. 9. In Fig. 13(a), we show the production cross sections for the scalar for various top partner masses and λ 2 = 1. The scalar-Higgs and top partner-top mixing angles are set to zero. In this limit, only the top partner loops contribute and for m T m S the cross section scales as ∼ λ 2 2 /m 2 T . Hence, the cross sections for different couplings and top partner masses can be easily obtained by rescaling these results. The cross sections for scalar production are found by rescaling the N 3 LO scalar gluon fusion production cross sections [129]. That is, we use the relevant Wilson coefficient for the g − g − S contact interaction for m T m S . With sin θ L = 0 and no Higgs-scalar mixing, S can decay into gg, γγ, γZ and ZZ through top partner loops, as shown in Fig. 9. We show the branching ratios of the scalar S in this limit in Fig. 13(b). For m T m S , all partial widths are proportional to λ 2 2 /m 2 T . Hence, the branching ratios are independent of the Yukawa coupling λ 2 and the top partner mass, and are determined by ratios of gauge couplings. Due to the strong coupling, the dominant decay mode is into gluons with BR(S → gg) 99%.

Experimental constraints
Some of the most constraining limits on colored particles come from QCD pair production shown in Figs. 2(a)-2(c). The production is mediated by the strong force, and the rate is completely determined by the mass, spin, and color representation of the produced particles. Hence, it is relatively model independent. There have been many searches for top pair production, but their applicability depends on on the precise decay pattern of the top partner. We summarize limits from pair production according to the mass and mixing categories in Table 2: • m S > m T − m t and sin θ L ∼ 0.1: The T → tS channel is forbidden, and the classic tree level decays T → th, T → tZ, and T → bW obey the expected relation BR(T → Figure 13. • m S < m T − m t and sin θ L ∼ 0.1: All tree level decays are available: T → tS, T → th, T → tZ, and T → bW . The traditional searches for pair produced top partners T → th, T → tZ, and T → bW [69][70][71][72] are then applicable. However, the branching ratios to th, tZ, and bW do not obey the expected pattern BR(T → bW ) ≈ 2 BR(T → tZ) ≈ 2 BR(T → th) ≈ 0.5, as shown in Table 2 and Fig. 8. Hence, the bounds are weakened. To fill in the gaps, searches for T → tS will have to be performed [58]. These will depend on the decay pattern of the scalar S, as discussed in Sections 3.2.1 and 4.
• m S > m T − m t and sin θ L ∼ 0: All tree level decays are very suppressed, and the loop level decays are relevant: T → tg, T → tγ, and T → tZ. A recent CMS analysis [78] searched for pair-produced spin 3/2 vector-like excited quarks T 3/2 which exclusively decays as T 3/2 → tg. The lower limit on the mass was found to be ∼ 1.2 TeV. While BR(T → tg) ∼ 1, the pair production rate of T is different from T 3/2 since T is spin 1/2. We recast the CMS search [78] to assess the current constraint on T using NNLO pair production cross section [72,[132][133][134][135][136]. The mass bound of this search is then m T 930 GeV. • m S < m T − m t and sin θ L ∼ 0: The decay channel T → tS dominates with branching ratio BR(T → tS) ∼ 1. This decay channel will require new search strategies [58], which will depend on the decay pattern of the scalar S and whether or not it mixes with the Higgs boson. See Sections 3.2.1 and 4 for a discussion.
To be conservative, we will assume the strongest constraints from pair production and work in the regime m T 1.2 − 1. 3 TeV. An alternative avenue to look for T in the high mass region is the EW single production in association with jets or W [16][17][18], as shown in Figs. 2(d)-2(e). Searches for the single production of T in ATLAS [137] and CMS [138,139] have excluded m T 1 − 1.8 TeV depending on the coupling strengths as well as branching ratios. For the SU (2) L singlet top partner model, this production mechanism vanishes as sin θ L → 0, and the constraints can be avoided.
Recent scalar resonance searches at the LHC in the gg [141], γγ [130,131], γZ [142] and ZZ [143] channels can put significant constraints on the scalar mass and couplings. Despite the small branching ratio, the S → γγ decay channel (BR 0.4%) is the cleanest, setting the most stringent limit on S. The experimental results are given for a low mass region 70 GeV < m S < 110 GeV [130] and a high mass region 200 GeV < m S [131]. Figure 14 demonstrates the excluded regions of the parameter space in the (a,c) lower and (b,d) higher m S regions, assuming for (a,b) m T = 1.5 TeV and (c,d) m T = 2.0 TeV. Scalar-Higgs and top partner-top mixing angles have been set to zero. The regions above these lines are excluded at the 13 TeV LHC. We show results for the (black solid) current data, and projections to (blue dash) 300 fb −1 and (red dot) 3 ab −1 . We have assumed both systematic and statistical uncertainties scale as the square root of luminosity. The outlook for the projected limits at the high luminosity-LHC with 3 ab −1 indicates that λ 2 is expected to be highly constrained λ 2 1 for the scalar S mass of ∼ 100 − 1000 GeV. The bound can be relaxed as the top partner mass increases, since the cross section decreases as 1/m 2 T .

Signal Sensitivity at the High Luminosity-LHC
The loop-induced single T production in association with a top quark, as shown in Fig. 4(b)-4(i), provides an unique event topology, offering useful handles to suppress the SM backgrounds. In this section, we present a detailed collider analysis for the high luminosity-LHC at √ S = 14 TeV with 3 ab −1 of data, and estimate the sensitivity reach in the final state We focus on the sin θ L = 0 limit so that BR(T → tS) ≈ 1 and BR(S → gg) ≈ 1. Both ggand qq-initiated processes are taken into account in the analysis. 3 We focus on the semileptonic decay of the tt system in order to evade a contamination from the QCD multi-jet background.

Signal Generation
To generate signal events described in Eq. (6.1), we first implement the EFT in Eq. (2.22) within the MadGraph5 aMC@NLO [144] framework using FeynRules [87,88]. The vertices needed for T and S decays can be conveniently parametrized by the interaction Lagrangian 4 and the effective operator We use the default NNPDF2.3QED parton distribution function [90] with fixed factorization and renormalization scales set to m T + m t . At generation level, we require all partons to pass cuts of p T > 30 GeV, and |η| < 5, where H T denotes the scalar sum of the transverse momenta of all final state particles. We will consider m S = 110 GeV, m T = 1.5 TeV and 2 TeV, and sin θ L = 0. The sin θ L = 0 limit is particularly interesting in this model because the production and decay patterns of the top partner are different from the the traditional approaches, as discussed in Section 3. We use such a small scalar mass so that the production cross section is maximized, as shown in Fig. 6(b). However, the EFT in Eq. (2.22) is not valid. Thus, we reweight the matrix element of the EFT by the exact one-loop calculation on an eventby-event basis. We also reweight the events according to the exact branching ratios of the decays T → tS and S → gg. Details of the T production and decay calculation are given in Sections 3.1 and 3.2, respectively, as-well-as the Appendices A and B. Details of the scalar decay can be found in Section 4. The reweighted events are showered and hadronized by PYTHIA6 [145] and clustered by the FastJet [146] implementation of the anti-k T algorithm [147] with a fixed cone size of r = 0.4 (1.0) for a slim (fat) jet. We include simplistic detector effects based on the ATLAS detector performances [148], and smear momenta and energies of reconstructed jets and leptons according to the value of their energies (see the details in Appendix C).  Table 4. The summary of the SM backgrounds after generation level cuts Eqs. (6.4-6.6). Matching refers to the either the 4-flavor or 5-flavor MLM matching. σ · BR denotes the production cross section (fb) times branching ratios including the top, W , and Z decays.

Background Generation
The SM backgrounds are generated by MadGraph5 aMC@NLO at leading order accuracy in QCD at √ S = 14 TeV with the NNPDF2.3QED parton distribution function [90]. All events are subject to the cuts in Eqs. (6.4-6.6). We use the default variable renormalization and factorization scales. The MLM-matching [149] scheme is used. The matching scales are chosen to be xqcut = 30 GeV and Qcut = 30 GeV for all backgrounds.
The most significant (irreducible) background is semi-leptonic tt + jets matched up to two additional jets. The relevant EW produced single-top backgrounds are tW and tq, where q is a light or b-quark. The tW -channel is generated with up to three additional jets and one W decays leptonically while the other decays hadronically. The tq channel is generated with up to two additional jets and we only consider a top quark which decays leptonically. Another relevant background includes W + jets with up to four additional jets and we only include a leptonically decaying W . Much smaller backgrounds include W W + jets with up to three additional jets where one W decays leptonically and the other hadronically. Finally, W Z + jets sample is generated with up to three additional jets where the W is forced to decay leptonically and the Z hadronically. Although W W and W Z are small compared to the other backgrounds, they are still large compared to the signal. A detailed summary of the backgrounds, the matching schemes, and their cross section after generation level cuts in Eqs. (6.4-6.6) is presented in Table. 4. It should be noted that tW is the dominant contribution to single top, whereas Fig. 3 would seem to indicate that tq should be dominant. However, while EW tq is dominant before cuts, the H T cut in Eq. (6.6) greatly reduces tq and tW becomes the leading contribution.
All background events are fed into PYTHIA6 [145] for parton showering and hadronization, and then clustered by the FastJet [146] implementation of the anti-k T algorithm [147]. We use two cone sizes of r = 0.4 and 1.0 for slim and fat jets, respectively. Momenta and energies of reconstructed jets and leptons are smeared in the same way of the signal 4 The kinematic distributions of final state particles can be sensitive to the chiral structure of the coupling t − T − S, since the polarization of the top quark propagates to daughter particles. Realizing sophisticated analysis to reflect all shapes of kinematic distributions is beyond the scope of our work. Here we will assume the relative size of the couplings is the same λ S tT = λ S T t .

Signal Selection and Sensitivity
Since we work in the parameter region that m T m S , m t , the top quark and S arising from the heavy T decay are kinematically boosted with high p T . Hence their decay products are highly collimated. To illustrate this, in Fig. 15(a) we show ∆R gg between the two gluons from the S decay, and in Fig. 15(b) we show ∆R W b between the b-quark and W from the top quark decay originating from the T . These plots are at partonic level before showering, hadronization, or detector effects have been considered. The angular separation ∆R ij is defined as where ∆φ ij = φ i − φ j is the difference of the azimuthal angles of particles i, j, and ∆η ij = η i −η j is the difference of the rapidities of the particle i, j. As can be seen, the distributions of ∆R gg and ∆R W b peak at ∆R gg ∼ ∆R W b ∼ 0.2 − 0.4. The other top quark produced together with T can also acquire a sizable p T , as shown in Figs. 15(c). This can be understood via Fig. 15(d), where we show the top partnertop invariant mass m T t distribution at partonic level. In the sin θ L = 0 limit, only loops containing top partners contribute to pp → T t. When m T t ∼ 2m T , the internal top partners can go on-shell, giving rise to the peaks in the m T t distributions. These peaks are quite pronounced. Hence, there is a relatively strong Jacobian peak at p T ∼ m T , causing the shoulder features in Fig. 15(c). 5 Since both tops and scalar are all boosted, we require that after showering, hadronization, and detector effects are accounted for that events contain at least one r = 1.0 fat jet with p j T > 400 GeV and |η j | < 2.5. (6.8) The variable r describes the cone-size of the anti-k T clustering algorithm [147], as described in Sections 6.1 and 6.2. Additionally, our signal consists of one leptonically decaying top t → b ν. Hence, we require that our events have missing transverse energy / E T > 20 GeV, (6.9) at least one r = 0.4 slim jet with p j T > 30 GeV and |η j | < 2.5, (6.10) and exactly one isolated lepton passing the cuts in Eq. (6.5) and mini − iso > 0.7. (6.11) The mini-iso [152] observable is defined as p T of a lepton divided by the total scalar sum of all charged particles' transverse energy (including the lepton) with p T > 1 GeV in the cone of radius ∆R = 10 GeV/p T . Since both tops are highly boosted the signal contains a fat jet originating from a top quark. Additionally, we have a fat jet originating from the decay of the scalar. Both these fat jets will have unique internal substructures due to the daughter particles. Such events are rare in the SM, and therefore serve as good handles to disentangle the SM backgrounds from our signal events. We use the TemplateTagger v.1.0 [153] implementation of the Template Overlap Method (TOM) [154,155] to tag massive boosted objects 6 . The TOM is based on an overlap Ov a i , where a is a parent particle and i is the number of daughter particles inside a fat jet. The closer Ov a i is to one, the more likely that a fat jet originated from the particle a. This method is flexible enough to tag any type of heavy object and is weakly susceptible to pileup contamination [155]. A multi-dimensional TOM analysis [20,157] extends its capability to further unravel multiple boosted objects with different internal substructures, and significantly improves a net tagging efficiency of the hadronically-decaying top and scalar S(→ gg) jets in the same event. For a precise definition see Refs. [153][154][155]. For a r = 1.0 fat jet to be tagged as the hadronic top, we demand a three-pronged top template overlap score Ov t 3 > 0.6. (6.12) We define a fat jet to be an S-candidate if it passes a two-pronged S template overlap score and is not tagged as a top-fat jet: Ov S 2 > 0.5 and Ov t 3 < 0.6. Finally, since the top-tagged fat jet should contain a b-quark, at least one b-tagged slim r = 0.4 jet should be found inside the top-tagged fat jet. 7 We require that exactly one 6 For alternatives to the TOM see Ref. [156] and references therein. 7 In our semi-realistic approach for the b-jet identification, r = 0.4 jets are classified into three categories where our heavy-flavor tagging algorithm iterates over all jets that are matched to b-hadrons or c-hadrons. If a b-hadron (c-hadron) is found inside, it is classified as a b-jet (c-jet). The remaining unmatched jets are called light-jets. Each jet candidate is further multiplied by a tag-rate [158], where we apply a flat b-tag rate of b→b = 0.7 and a mis-tag rate that a c-jet (light-jet) is misidentified as a b-jet of c→b = 0.2 ( j→b = 0.01). For a r = 1.0 fat jet to be b-tagged, on the other hand, we require that a b-tagged r = 0.4 jet is found inside a fat jet. To take into account the case where more than one b-jet might land inside a fat jet, we reweight a b-tagging efficiency depending on a b-tagging scheme described in Ref. [20]. top-tagged fat jet passes the cut in Eq. (6.14) and has a b-tagged slim jet inside, and exactly one scalar-tagged fat jet passes the cuts in Eqs. (6.15) and (6.16): N 1.5 t had = 1 and N 1.5 S = 1, respectively. (6.17) Table 5 is a cut-flow table showing the cumulative effects of cuts on signal and background rates. Relative to the basic cuts in Eqs. (6.8-6.11), under the requirement that N 1.5 t had = N 1.5 S = 1, the signal efficiency is 5.8%, while the major backgrounds tt and single t have efficiencies of 0.085% and 0.057%, respectively. The W and V V backgrounds are cut down to 0.0036% and 0.0028%, respectively, greatly diminishing the overall size of backgrounds.
For m T = 2 TeV, the reconstructed invariant mass and transverse momentum distributions for the top-tagged and scalar-tagged fat jets are shown in Fig. 17. The observations For the scalar-tagged fat jet we use the same mass window as Eq. (6.15), but harden the transverse momentum cut: p reco T,S > 560 GeV (6.19) However, as the T mass scale increases, we confront the challenge that the signal cross section steeply decreases, weakening our significance. To retain more signal events, we do not require that a b-tagged slim jet be found inside the top-tagged fat jet. Hence, for m T = 2 TeV we require exactly one top-tagged fat jet that passes the cut in Eq. (6.18) and without the b-tagging requirement, and exactly one scalar-tagged fat jet that passes the cuts in Eqs. (6.15) and (6.19): N 2.0 t had = 1 and N 2.0 S = 1, respectively. (6.20) As can be seen in Table 5, due to the relaxation of the b-tagging requirement, all efficiencies for background and signal are larger as compared to the m T = 1.5 TeV case. However, the backgrounds are still efficiently suppressed, especially the backgrounds that do not contain top quarks.
To further separate signal from background, it is useful to fully reconstruct the event. However, this means reconstructing the leptonically decaying top, t lep , and the missing neutrino momentum. First, to help reconstruct the top quark, we require that at least one of the slim jets passing the cuts in Eq. (6.10) is also tagged as a b-jet and meets the endpoint criteria where m b is the invariant mass of the b-tagged slim jet and isolated lepton, and Γ is a headroom to take into account effects of parton showering and hadronization. We choose Γ = 20 GeV to keep signal events up to ∼ 90%. We then reconstruct the momentum of the missing neutrino following the prescription in Ref. [159,160]. The total transverse momentum of the system is zero, so the transverse momentum of neutrino is just the missing transverse momentum. However, the longitudinal component of the neutrino momentum is still unknown and cannot be determined via momentum conservation since the longitudinal momentum of the initial state is unknown at hadron colliders. We will use the on-shell mass constraints that the invariant mass of the neutrino is p 2 ν = 0 and the invariant mass of the isolated lepton and neutrino satisfy m 2 ν = m 2 W . Since these are quadratic equations, there are two possible solutions for the neutrino longitudinal momentum where A = m 2 W + 2 p T · / E T , p L is the lepton longitudinal momentum, p is the lepton's three-momentum, p T is the lepton's transverse momentum vector, and / E T is the missing transverse energy vector. To break the two fold-ambiguity of Eq. (6.22) and to determine which b-jet originates from the leptonically decay top, we use the top quark mass constraint. We select the b-jet and p ν L pair that minimizes the quantity where m b ν is the invariant mass of a b-jet, lepton, and neutrino system. The resulting b-jet and neutrino momentum are used to reconstruct the leptonically decaying top, t lep . Once we have reconstructed the leptonically decay top, we require that it has the correct mass and has fairly high p T : 150 GeV < m reco t lep < 210 GeV, p reco T,t lep > 500 GeV, for m T = 1.5 TeV, and (6.24) 150 GeV < m reco t lep < 220 GeV, p reco T,t lep > 680 GeV, for m T = 2 TeV. (6.25) As we can see from the fourth rows of Table 5, as compared to the N t had and N S cuts, after t lep reconstruction the vector boson backgrounds are reduced by 2 − 5 orders of magnitude, the single top background efficiency is 1 − 4%, and the tt efficiency is 1 − 7%. The signal efficiency is 20 − 40%. Although the background is greatly reduced, to suppress it further relative to signal we will use the reconstructed top partner mass m reco T and the total invariant mass of the reconstructed system. While the top quarks and scalar are fully reconstructed, it is not clear yet which top quark originated from the top partner decay. We select the pair {S, t i }, where i = had, lep denotes either the hadronic or leptonic top, that best reconstructs T by minimizing the mass asymmetry variable The effects of the the cuts in Eqs. (6.27-6.30) on signal and background are shown in the fifth and sixth rows of Table 5. After these cuts the background and signal rates are comparable. However, one final set of cuts is made to increase the significance of the signal. We introduce a variable H reco As shown in the last row of the tables in Table 5, these final cuts decrease the background cross section to below the signal rate, with good signal efficiency.
To quantify the observability of our signal at the LHC, we compute a significance (σ) using the likelihood-ratio method [161] σ ≡ −2 ln L(B|Sig+B) where Sig and B are the expected number of signal and background events, respectively. All significances in Table 5 are calculated for given luminosity of 3 ab −1 and given in the last row. While the cuts from the basic to the reconstructed invariant mass m reco T t in Eqs. (6.8-6.30) decrease background rates until they are comparable to signal, it is the final cuts in Eqs. (6.32) and (6.33) that significantly increase the significance. The final signal significance turns out to be 5.0 for the benchmark parameter point m T = 1.5 TeV, λ 1,2 = 2, m S = 110 GeV and sin θ L = 0 assuming a luminosity of 3 ab −1 . Although we can achieve the high significance, only ∼ 1.7 signal events are expected. While this may be enough to set constraints on the model, it is not enough for discovery. The sensitivity for heavier T mass scales become weaker, where the final signal significance turns out to be 1.9 for the benchmark parameter point m T = 2.0 TeV, λ 1,2 = 3, m S = 110 GeV and   A summary of the backgrounds can be found in Table 4. sin θ L = 0 with the same amount of the luminosity. However, due to relaxation of b-tagging inside the hadronically decay top quark fat jet, we actually expect 2.4 signal events, more than m T = 1.5 TeV.
From these results, we can project sensitivities for many coupling constants. For sin θ L = 0, the production cross section is proportional to λ 2 1 λ 2 2 , Eq. (3.1). Additionally, the branching ratio of T → tS is essentially one. Hence, we can simply scale the signal cross sections in Table 5 to determine significances for different coupling constants. In Fig. 21, we summarize the final significance contours for two benchmark T masses (a) m T = 1. confidence level, using this channel the LHC will be able to exclude

Conclusions
We have studied a simple extension of the SM with a SU (2) L singlet fermionic top partner and gauge singlet scalar. These top partners are ubiquitous in composite completions of the SM, and are needed to help make the Higgs natural. Additionally, singlet scalars are present in many SM extensions and can provide a useful laboratory to categorize new physics signatures at the LHC. While there have been many studies and searches for top partners, this model presents a unique phenomenology with many interesting characteristics. At tree level, if the new scalar is light enough, the top partner has a new decay channel T → tS that can have a large branching ratio and will require new search strategies at the LHC [58]. In particular, if the mixing angle between the top partner and top quark vanishes sin θ L = 0, then BR(T → tS) ≈ 1, when it is kinematically allowed, as discussed in Section 3.2.1.
However, the precise decay channel of the scalar S will depend on its mixing with the Higgs boson. If that mixing is non-negligible, we can expect S to decay much like a heavy Higgs, with the additional S → hh decay channel. If the scalar-Higgs mixing is zero, S will predominantly decay to gluon S → gg through top-partner loops.
Of particular interest to us, this model introduces many new loop induced production and decay modes of the top partner. It is possible to produce the top partner in association with the top quark (T t) through loops as shown in Fig. 4. For the singlet top partner, the typical production mode is T T or single top partner production in association with a jet or W . These single top partner production modes depend on the T − b − W or T − t − Z couplings, which are suppressed by the top partner-top mixing angle (Eq. (2.19)). In the limit that this mixing angle goes to zero, these production modes vanish. However, the loop induced diagrams for T t production persist. As the LHC quickly saturates the phase space needed to pair produce the top partner, the T t channel will become increasingly important. In fact, we found that for reasonable coupling constants, the T t production rate can overcome the T T production rate for top partner masses of m T 1.5 TeV, as discussed in Section 3.1. Our results for top partner production are summarized in Table 1.
Loop induced decays can also be quite interesting. For non-negligible top partner-top mixing, the traditional decay modes T → tZ, T → th, and T → bW dominate. However, similar to single top partner production, these decay modes vanish as top partner-top mixing vanishes. In this limit, the scalar can mediate loop-induced the decay channels T → tg, T → tγ, and T → tZ, through the loops shown in Figs. 7(e)-7(g). These loops do not vanish is the small mixing limit. When sin θ L = 0 and m S > m T , these decays dominate. Since the loops are all of a similar form, the branching ratios are determined by the gauge couplings and T → tg is the main decay mode. While these decay channels have been searched for [78], in this model they are loop induced and the top partner can be quite long lived, as discussed in Section 3.2.2. In fact, for most of the parameter range the top partner hadronizes before it decays. For not too small couplings, it is possible to search for displaced vertices, "stable" particles, and stopped particles. Our results for top partner decays are summarized in Tables 2 and 3.
Whether T → tS dominates, T → tg dominates, or the top partner hadronizes and is long lived, new search strategies are needed at the LHC to fully probe the parameter space of this model. To this end we have performed a collider study focusing on the exotic production mode pp → T t + tT . We focused on the small scalar mass case, in order to maximize the production rate, as shown in Fig. 6(b). We also focused on sin θ L = 0, so that other single top partner modes decouple and the exotic T → tS → tgg decay mode dominates. This mode provided many boosted particles, allowing us to get a good handle on the signal. This is a new production mode that provides an exotic signature at the LHC. With 3 ab −1 of data, we found that this production and decay mode can probe much of the parameter space inaccessible to other processes, as shown in Fig. 21.
As the LHC continues to gain data and new physics continues to remain elusive, it becomes imperative that we leave no rock unturned. This means we must go beyond the simplest simplified models and search for new signals. The model presented in this paper provides many new signatures of top partners that have not yet been searched for. These included promptly decaying top partners with new decay channels, long live top partners with exotic decay channels, and new production channels for single top partner production.
In much of the parameter space, these signatures are available with reasonable masses and coupling constants. hospitality and its partial support during the completion of this work. JHK is grateful to Tae Hyun Jung for valuable help and discussions. We also thank the HTCaaS group of the Korea Institute of Science and Technology Information (KISTI) for providing the necessary computing resources. This work is supported in part by United States Department of Energy grant number DE-SC0017988 by the University of Kansas General Research Fund allocation 2302091. The data to reproduce the plots has been uploaded with the arXiv submission or is available upon request.

A Wavefunction and Mass Renormalization of Top Partner
We renormalize the bare Lagrangian of the top sector based on the on-shell wave function renormalization scheme [162][163][164][165], largely following the method of Ref. [162]. We start with the bare kinetic and mass terms of the top quark and top partner after electroweak symmetry breaking and mass diagonalization: where the superscript 0 indicates bare quantities. We allow for different wave-function renormalization constants for left-and right-handed fields, as well as for ψ and ψ: where τ = L, R, Z τ ij and Z τ ij are renormalization constants, δZ τ ij and δZ τ ij are counterterms (CTs), and fields without the 0 subscript are the physical, renormalized fields. We renormalize the masses via To determine the wavefunction and mass CTs, we start with the two-point Feynman rules for the CTs at one-loop where the momentum p is moving to the left with particle flow. To calculate renormalization constants, we consider a propagator which mixes different families through radiative corrections whereΣ ij ( / p) is a renormalized self-energy decomposed into all possible Dirac structureŝ and Σ ij ( / p) is the one-loop one-particle irreducible unrenormalized two point function: Off diagonal wave function renormalization constants can be obtained by using the renormalization conditions that i − j mixing vanishes when either i or j are on-shell: where Re indicates that the real and complex pieces of the coupling constants are retained, but the absorptive pieces of the loop integrals are dropped [166]. The off-diagonal wavefunction renormalization constants are then [162] δZ Now we turn to the diagonal entries of the propagator Eq.(A.5). We impose three conditions [162], two of which are the normal pole and residue constraints. These conditions are imposed after explicitly inverting S −1 ii .
1. The numerator of S ii should not be chiral when the particle is on-shell p 2 = m 2 i .
2. The propagator S ii should have a pole at p 2 = m 2 i .
3. When on-shell, the propagator should have unit residue: See Ref. [162] for details of the calculation. For completeness, we summarize their results here: 12) and the primes indicate derivative with respect to the argument p 2 . The α i are arbitrary constants that reflect that there are not enough renormalization conditions to fully determine the wavefunction and mass CTs. We will choose α i = 0.

A.1 Off-diagonal Mass Counterterms
When renormalizing, it is possible to have off-diagonal mass CTs as well as the diagonal CTs in Eq. (A.3). Some literature includes the off-diagonal CTs [163][164][165], while others do not [162]. The two approaches are equivalent, and it is a choice whether or not to include them. This is because the off-diagonal renormalization conditions in Eq. (A.8) are insufficient to uniquely solve for both the off-diagonal wave-functions CTs and the off-diagonal mass CTs. We start by adding off-diagonal mass CTs, and will assume all mass terms are real. Tildes indicate fields in the non-zero mass CT scheme. After mass renormalization, but before wave function renormalization, the mass terms are The hermiticity of the mass terms requires that δm L tT = δm R T t and δm L T t = δm R tT . (A.14) These mass terms can be diagonalized via the usual bi-unitary transformation where δZ τ is the wavefunction CT matrix in Eq. (A.2) and δ Z τ is an equivalent wavefunction CT matrix with non-zero mass CTs. We have used the fact that after full renormalization the schemes with and without the off-diagonal counterterms have to produce the same renormalized physical fields. That is, whether we diagonalize the mass CT matrix then perform wave-function renormalization or perform wave-function renormalization and find a scheme to determine the off-diagonal mass CTs, the final renormalized fields should be the same. Hence, on the right-hand-side of Eq. (A.18), the final renormalized fields are the same. We can then read off the relationship between the wave-function CTs with nonzero or zero mass CTs: Similarly, the relationship for the renormalization of the barred fields is

B Vertex Counterterms and Mixing Angle Renormalization
We now turn to renormalization of the interactions between the top partner and top quark. The only interactions that we consider at one-loop and are T − t − g, T − t − γ, and T − t − Z. These have the added complication that flavor changing interactions need to be renormalized, including quark mixing [162-165, 167, 168].
Since there are no tree-level interactions between T − t − g and T − t − γ, the vertex counterterms originate from wavefunction renormalization. For T − t − g the, the counterterms in Sec. A are sufficient. For the T − t − γ interaction, the Z − γ wavefunction renormalization must also be considered. Following [169], the counterterms are where, again, the superscript 0 indicates unrenormalized quantities. To find ∆ 0 and δZ Zγ , construct the renormalized two-point function (B.5) For T − t − Z we need mixing angle and coupling constant renormalization as-well-as wave-function CTs. The wave-function renormalization δZ Z can be determined by the usual requirements that the Z-propagator has a pole at p 2 = m 2 Z and that it has unit residue: where Π 0 ZZ (p 2 ) is an unrenormalized two-point function defined similarly to Π 0 γZ (p 2 ) in Eq. B.3. The coupling constant and mixing angle CTs are defined as where t W = s W /c W . We refer the reader to Ref. [169] for details on calculating δe and δs W . The relevant vertex counterterms are then The final piece needed in the mixing angle CT, δθ L . We focus on renormalizing T → tZ. We then calculate T → tZ and determine δg L in the MS scheme. At one loop for T → tZ diagrams with the scalar S, Higgs, Goldstones, W , and Z are included. In this way, all corrections from Yukawa couplings are included in a gauge invariant way. Diagrams with gluons are not included, since they are corrections to the tree level T → tZ, and so vanish as the tree level T → tZ vanishes. In addition, gluons have a separate gauge parameter from the EW sector and are not needed for gauge invariance. We have verified that in the limit m t , m Z , sin θ L → 0 limit that the Lorentz structure of the EFT in Eq. (2.22) is recovered.

C Parameterization of Detector Resolution Effects
We include detector effects based on the ATLAS detector performances [148]. The jet energy resolution is parametrized by noise (N ), stochastic (S), and constant (C) terms where in our analysis we use N = 5. where σ ID = E a 2 1 + (a 2 E) 2 (C.3) We use a 1 = 0.023035, a 2 = 0.000347, b 0 = 0.12, b 1 = 0.03278 and b 2 = 0.00014 in our study.